The popularity of the phrase 'roll in the hay' in the 1940s, such as it was, was perhaps attributable to the action in a 1941 book made into an Academy Award-winning movie: This Above All, by Eric Knight.
At first, I was unconvinced that the phrase enjoyed any special popularity in the '40s. Google Ngrams shows a different picture.
The occurrences linked below the Ngram also do not bear out the claim: 1800-1942 gives only two references, a 1934 book of that title described as "English wit and humour", and an 1810 song book; 1943-1990 gives only five references, including another book titled A Roll in the Hay, this one from 1979 and described as "Paraphilias".
As usual, however, Google Ngrams proved deceptive. COHA for the phrase 'in the hay' showed two usage bumps in that corpus, in the 1920s and 1940s:
Examining the context of the uses in the 1920s and 1940s, it can be seen that 'in the hay', while instanced twice with a prefaced roll in 1941, is used with other senses and constructions in the 1920s. The two exact instances of the phrase in 1941 both refer to sex:
She looked at me and I saw how beautiful her eyes could be when they weren't being hard and I met the look, inquisitively, and she turned away to stare out at the lights far below. " Hell, this isn't a roll in the hay for me, there's plenty of that around. This is, well, maybe the best way of saying good-bye... "
(From What Makes Sammy Run?, Budd Schulberg, 1941.)
"Not a place for picnicking, or a roll in the hay," he said.
(From Mildred Pierce, James Cain, 1941.)
Note that none of the 12 instances from the 1930s used the exact phrase 'roll in the hay'; one of those instances was, however, synonymous with 'roll in the hay': 'sported together in the hay'.
Of the 11 instances in the 1950s, five are in the sense of 'sex in the hay':
... You and me would be terrific in the hay, baby.
(From Return to Paradise, James A. Michener, 1951.)
If he should end up in the hay with Angela tonight ...
(From Rally Round the Flag, Boys!, Max Schulman, 1957.)
They gave me a gold star or a roll in the hay with the principal's wife or something.
(From Plowshare in Heaven, Jesse Stuart, 1958.)
... if you stayed on you'd wind up in the hay with Mama.
(From Hawaii, James A. Michener, 1958.)
... I did not wind up in the hay with Mama.
From these data, which show three instances of use of the phrase 'in the hay' in This Above All in 1941, and acknowledging the popular appeal of This Above All, both the book and the 1942 movie, with its plot-line that includes a central character, Clive, who is disgusted enough by the war to go AWOL,
having a crisis of conscience over what the war is being fought for and disgusted at the incompetence of the ruling elite, Clive decides not to return to the Army and to go absent without leave,
(Wikipedia, "This Above All")
and so as a sop for the US people's ambivalence about US involvement in the war at the time,
a large majority of the American public continued to oppose any direct military intervention into the conflict well into 1941,
(Wikipedia, "World War II")
the reason for the popularity of the phrase in the 1940s may be explained. The exact phrase 'roll in the hay' did not, so far as I know, occur in the book or the movie; the action of the book and the movie both, though, perhaps contributed to the adoption of the phrase (and particularly Michener's repeated use of it in the 1950s) with the particular sense of, if not casual, at least carefree sex.
Other similar expressions are numerous. I'll list a few I've heard or seen in use, but an exhaustive listing is quite impossible. For more, beyond my limited experience, see for example the list at The Online Slang Dictionary.
- a bit of how's your father,
- beat cheeks,
- bump uglies,
- dip [your] wick,
- do the horizontal bop,
- do the nasty,
- get down and dirty,
- get laid,
- get lucky,
- go at it,
- make woopie,
- slap and tickle.